THE DEFICIENCY MODULE OF A CURVE AND ITS ... - CoCoA System

THE DEFICIENCY MODULE OF A CURVE AND ITS SUBMODULES GUNTRAM HAINKE AND ALMAR KAID

1. Introduction This is a summary of the fifth tutorial handed out at the CoCoA summer school 2005. We discuss the so-called deficiency module for projective curves. In particular, we provide the necessary CoCoA (version 4.6) codes for implementation. We thank Holger Brenner, Martin Kreuzer and Juan Migliore for helpful discussions. Let K be a field and P := K[x0 , . . . , xn ] be a polynomial algebra equipped with the standard grading. By m := (x0 , . . . , xn ) we denote the graded maximal ideal. Moreover, by IC ⊂ P we denote a homogeneous ideal defining a curve C ⊂ Pn = PnK = Proj P and let R := P/IC be the coordinate ring of C. Throughout this article we assume that all ideals defining a projective subscheme are saturated, i.e. I = I sat = {f ∈ P |mt · f ⊂ I for some t > 0}. This can be checked with CoCoA, for example, with the function Define IsSaturated(I) M:=Ideal(Indets()); Return Saturation(I,M)=I; EndDefine; Definition 1.1. The deficiency module or Hartshorne-Rao module of a curve C ⊂ Pn is the graded P -module M (C) :=

M

H 1 (Pn , IC (t)),

t∈Z

where IC = If C is the ideal sheaf corresponding to IC . The deficiency module is also describable as the first local cohomology of R = P/IC . 1

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Proposition 1.2. Let C ⊂ Pn , n ≥ 2, be a curve with coordinate ring R = P/IC . Then M (C) = Hm1 (R). Proof. From the short exact sequence 0 −→ IC −→ P −→ R −→ 0 we derive the exact sequence in cohomology Hm1 (P ) −→ Hm1 (R) −→ Hm2 (IC ) −→ Hm2 (P ). Since n ≥ 2, we have Hm1 (P ) = 0 = Hm2 (P ) (cf. [2, Corollary A1.6]). Therefore, Hm1 (R) ∼ = Hm2 (IC ). Now the assertion follows, since Hm2 (IC ) =

M

H 1 (Pn , IC (t))

t∈Z

(cf. [2, Corollary A1.12(2)]).

¤

Definition 1.3. A curve C ⊂ Pn is called arithmetically Cohen-Macaulay if its coordinate ring R = P/IC is Cohen-Macaulay, i.e. dim R = depth R. We have the following characterization for arithmetic Cohen-Macaulay curves utilizing its deficiency module. Corollary 1.4. Let C ⊂ Pn , n ≥ 2, be a curve. Then C is arithmetically Cohen-Macaulay if and only if M (C) = 0. Proof. Let C be arithmetically Cohen-Macaulay. Then we have that Hmi (R) = 0 for all i < 2, since dim R = depth R = 2. Hence by Proposition 1.2 we have M (C) = Hm1 (R) = 0. On the other hand, supppose that M (C) = Hm1 (R) = 0. Because IC is saturated, Hm0 (R) = 0. Since dim R = 2, we have Hm2 (R) 6= 0 and Hmi (R) = 0 for all i > 2. Hence dim R = depth R, i.e. R is Cohen-Macaulay and therefore C is arithmetically Cohen-Macaulay. (For all the vanishing and nonvanishing statements cf. [2, Proposition A1.16].) ¤ By using Proposition 1.2 above we can express the deficiency module of a curve in terms of Ext-modules. Theorem 1.5. Let C ⊂ Pn , n ≥ 2, be a curve with coordinate ring R. Then the modules M (C) and HomK (ExtnP (R, P ), K)(n + 1) are isomorphic as graded P -modules. ∼ H 1 (R). The local duality Proof. By Proposition 1.2, we have M (C) = m Theorem (cf. [2, Theorem A1.9]) yields H 1 (R) ∼ = Extn (R, P (−n − 1))∗ = HomK (Extn (R, P ), K)(n + 1) m

P

P

THE DEFICIENCY MODULE OF A CURVE AND ITS SUBMODULES

and proves the claim.

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Theorem 1.5 allows a quick computation of the K-dual of M (C) with CoCoA, using the implemented Ext-package. This is convenient for most of our computations in this paper. Therefore, we define the CoCoA-function DeficiencyDual(I) which takes the vanishing ideal I of a curve C and computes a presentation of the Ext-module ExtnP (R, P ), where R is the coordinate ring of C: Define DeficiencyDual(I) N:=NumIndets()-1; Return Ext(N,CurrentRing()/I,Ideal(1)); EndDefine; We now turn to finding a presentation of the K-dual of a P -module M which is a finite-dimensional K−vector space. Here we follow essentially the explanations in [1]. So let M ∼ = P k /N be a presentation k of M and < a module term ordering on P . Then a K−vector space basis of M is given by B := Tn+1 he1 , . . . , ek i\ LT< (N ), where Tn+1 he1 , . . . , ek i denotes the set of terms in K[x0 , . . . , xn ]k and LT< (N ) the leading term module of N with respect to = 1 ------------------------------i.e. M (C1 )0 = K and M (C1 )t = 0 for t > 0. In particular C1 is not arithmetically Cohen-Macaulay. Example 2.2. Let C2 := V+ (x0 x2 − x21 , x1 x3 − x22 , x0 x3 − x1 x2 ) ⊂ P3 be the twisted cubic curve. So we use the commands Use P::=Q[x[0..3]]; IC_2:=Ideal(x[0]x[2]-x[1]^2,x[1]x[3]-x[2]^2, x[0]x[3]-x[1]x[2]); and compute the resolution of P/IC2 by Res(P/IC_2); CoCoA yields: 0 --> P^2(-3) --> P^3(-2) --> P ------------------------------We see that pd(P/IC2 ) = 2 = codim IC2 where codim I = min{ht(p) : I ⊆ p minimal}. Hence R = P/IC2 is Cohen-Macaulay and therefore by Corollary 1.4 we get M (C2 ) = 0. Example 2.3. Here we consider the smooth rational quartic curve C3 ⊂ P3 defined as the vanishing of the 2 × 2 minors of the matrix µ

x0 x21 x1 x3 x2 x1 x0 x2 x22 x3

¶

.

We realize the vanishing ideal of the curve C3 by the commands Use P::=Q[x[0..3]]; M:=Mat([[x[0],x[1]^2,x[1]x[3],x[2]], [x[1],x[0]x[2],x[2]^2,x[3]]]); IC_3:=Ideal(Minors(2,M)); Now we look at the K-dual of the deficiency module M (C3 ) by MC_3:=DeficiencyDual(IC_3); If we compute the Hilbert function of M (C3 ) via Hilbert(MC_3); we get H(0) = 1 H(t) = 0 for t >= 1 -------------------------------

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GUNTRAM HAINKE AND ALMAR KAID

Hence M (C3 ) is one-dimensional. Example 2.4. In this example we want to study the deficiency module of a curve C4 ⊂ P3 given as the disjoint union of a line and a plane curve of degree d for some d ∈ N. So we realize the situation (for d = 3) with CoCoA in the following way: Use P::=Q[x[0..3]]; IL:=Ideal([GenLinForm(10),GenLinForm(10)]); Use S::=Q[x[0..2]]; D:=3; F:=Randomized(DensePoly(D)); Use P; IPC:=Ideal(BringIn(F),x[3]); Now we apply again Proposition 1.7 and compute the deficiency module M (C4 ) via I:=IL+IPC; MC_4:=P/I; Now we compute the Hilbert function of M (C4 ) by Hilbert(MC_4); and get H(0) = 1 H(1) = 1 H(2) = 1 H(t) = 0 for t >= 3 ------------------------------i.e. in the case d = 3 we have dimK (M (C4 )) = 3. If we do this for various values d we get dimK (M (C4 )) = d as it should be. Example 2.5. We want to compute the deficiency module for a curve C5 ⊂ P3 given as the union of a line and a plane curve of degree d, d ∈ N, which meet in (at least) one point. To do this we fix the (intersection) point y := V+ (x1 , x2 , x3 ). So we compute the vanishing ideal of C5 (for d = 3) in the following way: Use P::=Q[x[0..3]]; G_1:=Sum([Rand(-10,10)*x[T]|T In 1..3]); G_2:=Sum([Rand(-10,10)*x[T]|T In 1..3]); IL:=Ideal(G_1,G_2);

THE DEFICIENCY MODULE OF A CURVE AND ITS SUBMODULES

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So IL is the ideal corresponding to a line in P3 passing through y. Now we construct an appropriate plane curve meeting the line L in the point y. Use S::=Q[x[0..2]]; D:=3; F:=Randomized(DensePoly(D)-x[0]^D); Use P; IPC:=Ideal(BringIn(F),x[3]); Hence we get the vanishing ideal of C5 by IC_5:=Intersection(IL,IPC); and obtain the K-dual of the deficiency module as usual by DeficiencyDual(IC_5); CoCoA yields: Module([0]) ------------------------------As in the previous example we get the same result for various values of d. So it is arithmetically Cohen-Macaulay. Example 2.6. We consider the coordinate cross in P3 defined by the intersection Icross of the ideals (x1 , x2 ), (x1 , x3 ) and (x2 , x3 ) in P = K[x0 , x1 , x2 , x3 ]. We realize I in CoCoA via Use P::=Q[x[0..3]]; L:=[Ideal(x[1],x[2]),Ideal(x[1],x[3]),Ideal(x[2],x[3])]; ICross:=IntersectionList(L); Now we compute a regular sequence of type (3, 3) in Icross which defines a complete intersection curve containing the coordinate cross. To do this we use the function GenRegSeq(I,L) defined in [3]: ICi:=Ideal(GenRegSeq(ICross,[3,3])); Now we look at the vanishing ideal IX of the residual curve X by IX:=ICi:ICross; If we compute the Hilbert polynomial of X via Hilbert(P/IX); we get HPX (t) = HPP/IX (t) = 6t − 2 for t ≥ 1, i.e. X is a curve of degree 6 and genus 3. Furthermore, by using DeficiencyDual(IX);

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GUNTRAM HAINKE AND ALMAR KAID

we notice that M (X) = 0, i.e. by Corollary 1.4 the curve X is arithmetically Cohen-Macaulay. We check with CoCoA that X is a smooth curve by computing the singular locus: Define IsSmooth(I) J:=Jacobian(Gens(I)); L:=List(Minors(1,J)); SingLoc:=Radical(Ideal(L)+I); Return SingLoc=Ideal(Indets()); EndDefine; IsSmooth(IX); Next, we construct another smooth curve in P3 of degree 6 and genus 3 which is not arithmetically Cohen-Macaulay. For this we go back to Example 2.1 and consider the curve C1 given as the union of two skew lines in P3 . We obtain the vanishing ideal of C1 by IC_1:=Intersection(IL_1,IL_2); We have already shown that C1 is not arithmetically Cohen-Macaulay and it is known that this is invariant under linkage. In the first step we link the curve C1 via a complete intersection curve of type (3, 4) to a curve X1 . To do this we use the commands: ICi_1:=Ideal(GenRegSeq(IC_1,[3,4])); IX_1:=ICi_1:IC_1; Now we link the curve X1 further via a complete intersection curve of type (4, 4) to a curve X2 : ICi_2:=Ideal(GenRegSeq(IX_1,[4,4])); IX_2:=ICi_2:IX_1; Now we check the Hilbert polynomial with Hilbert(P/IX_2); and get HPX2 (t) = HPP/IX2 (t) = 6t − 2 for t ≥ 1. And indeed, we use DeficiencyDual(IX_2); to check that M (X2 ) 6= 0, i.e. X2 is not arithmetically Cohen-Macaulay. Finally, the smoothness of X2 is established via IsSmooth(IX_2); 3. A degree bound for curves in P3 In this section we want to study a degree bound for curves in P3 given in [5] and compute some examples. In the sequel C denotes a

THE DEFICIENCY MODULE OF A CURVE AND ITS SUBMODULES

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curve in P3 . Firstly, we define a submodule of the deficiency module M (C) of C. Definition 3.1. Let `, `0 ∈ P1 be two general linear forms and let A := (`, `0 ) denote the ideal they generate. Then we define the P submodule KA ⊆ M (C) as the submodule annihilated by A, i.e. KA = 0 :M(C) A. We want to compute with CoCoA a presentation of the module KA . To do this, we need a presentation for the annihilator K` := 0 :M(C) (`) first, where ` ∈ P1 is a general linear form. So we write a function K_L(I), which computes a presentation of K` for a curve defined by the ideal I. Define K_L(I) MCdual:=DeficiencyDual(I); MC:=KDual(MCdual); GensMC:=Gens(MC); N:=Len(GensMC[1]); F:=GenLinForm(10); L:=Concat([F*E_(I,N) | I In 1..N],GensMC); S:=Syz(L); GensCol:=[Vector(First(List(T),N)) | T In Gens(S)]; Return Module(GensCol); EndDefine; With the help of the function K_L(I) above we can easily write a function K_A(I), which computes a presentation of the submodule KA ⊆ M (C) for a curve defined by the ideal I: Define K_A(I) KL_1:=K_L(I); KL_2:=K_L(I); Return(Intersection(KL_1,KL_2)); EndDefine; Definition 3.2. Let M be a finitely generated graded P -module with graded minimal free resolution 0 −→

bs M j=1

P (−ds,j ) −→ · · · −→

b0 M

P (−d0,j ) −→ M −→ 0.

j=1

We define mi (M ) := min{di,1 , . . . , di,bi }.

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GUNTRAM HAINKE AND ALMAR KAID

The following theorem (cf. [5, Corollary 2.3]), which provides a lower bound for the degree of a curve in P3 , builds the fundament for our further studies. Theorem 3.3. Let C ⊂ P3 be a curve. If m2 (P/IC ) ≤ m1 (KA ) + 2 then 1 deg(C) ≥ · m1 (P/IC ) · m2 (P/IC ) − dimK (KA ). 2 Example 3.4. We come back to Example 2.1 and consider the curve C1 given by the union of two skew lines in P3 . Since we have already shown that M (C1 ) = K 6= 0, we know that C1 is not arithmetically Cohen-Macaulay (cf. Corollary 1.4). Next we compute the vanishing ideal of C1 by IC_1:=Intersection(IL_1,IL_2); where IL_1 and IL_2 were defined in Example 2.1. Now we compute the Hilbert polynomial of the curve C1 utilising the command Hilbert(P/IC_1); and get H(0) = 1 H(t) = 2t + 2 for t >= 1 ------------------------------i.e the Hilbert polynomial of C1 equals HPC1 (t) = 2t + 2. Hence deg C1 = 2. Now we obtain the numbers m1 (P/IC1 ) and m2 (P/IC1 ) by computing the resolution of P/IC1 : Res(P/IC_1); The answer of CoCoA is 0 --> P(-4) --> P^4(-3) --> P^4(-2) --> P ------------------------------Thus we get m1 (P/IC1 ) = 2 and m2 (P/IC1 ) = 3. This example illustrates that the stronger degree bound 1 deg(C) ≥ · m1 (P/IC ) · m2 (P/IC ) 2 does not hold in general for curves which are not arithmetically CohenMacaulay, since for the curve C1 the right hand side equals 3. Example 3.5. We consider a curve C ⊂ P3 given as the disjoint union of two complete intersections of type (16, 16) (cf. also [5, Example 2.7]). To simplify matters we consider the complete intersections I1 := 16 16 16 (x16 0 , x1 ) and I2 := (x2 , x3 ). We realise the vanishing ideal of C via

THE DEFICIENCY MODULE OF A CURVE AND ITS SUBMODULES

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Use P::=Q[x[0..3]]; I_1:=Ideal(x[0]^16,x[1]^16); I_2:=Ideal(x[2]^16,x[3]^16); IC:=Intersection(I_1,I_2); We use Proposition 1.7 to compute the deficiency module M (C): I:=I_1+I_2; MC:=P/I; Since I1 + I2 is an m-primary ideal, the Hilbert function of M (C) = P/(I1 +I2 ) has only finitely many non-zero values. Therefore, we obtain the dimension of M (C) by Sum:=0; T:=0; While Hilbert(MC,T)0 Do Sum:=Hilbert(MC,T)+Sum; T:=T+1; EndWhile; Now the command Sum; yields dimK (M (C)) = 65536. We compute the Hilbert polynomial of our curve C as usual by Hilbert(P/IC); and verify that the degree of C is 512 = 2 · 162 as expected. Next we compute the minimal graded free resolutions of P/IC with Res(P/IC); and get 0 --> P(-64) --> P^4(-48) --> P^4(-32) --> P ------------------------------Hence m1 (P/IC ) = 32 and m2 (P/IC ) = 48. Now we compute a representation for the annihilator K` for a general linear form `. Since we have given the deficiency module M (C) = P/I as a quotient of the polynomial ring with I = I1 + I2 in this special example, we can compute it without using the Ext-package in the following way: We have K` = {(g + I) ∈ P/I : ` · (g + I) = 0 + I}. Therefore we have to compute the preimage of K` in the polynomial ring P which equals the ideal quotient J1 := I :P (`). To do this we use the commands L_1:=Ideal(GenLinForm(10)); J_1:=I:L_1; To get the dimension of K` as a K-vector space we compute at first via Len(QuotientBasis(J_1));

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GUNTRAM HAINKE AND ALMAR KAID

the lenght of a K-basis of P/J1 which equals 62800. Hence we have dimK (K` ) = 65536 − 62800 = 2736. In the same way we compute a representation of the submodule KA ⊆ M (C): L_2:=Ideal(GenLinForm(10),GenLinForm(10)); J_2:=I:L_2; The command Len(QuotientBasis(J_2)); yields dimK (P/J2 ) = 65365 and therefore we have dimK (KA ) = 65536 − 65365 = 171. Next we want to compute the value m1 (KA ), i.e. the minimal degree of a generator of KA . To do this we compute the graded minimal free resolution of P/I and P/J2 respectively by Res(P/I); Res(P/J_2); and get 0 --> P(-64) --> P^4(-48) --> P^6(-32) --> P^4(-16) --> P ------------------------------0 --> P^2(-63) --> P(-42)(+)P^2(-43)(+)P^8(-47)(+)P(-62) --> P^6(-32)(+)P^2(-41)(+)P^4(-42)(+)P^4(-46) --> P^4(-16)(+)P(-40)(+)P^2(-41) --> P ------------------------------Since KA ∼ = J2 /I we see that m1 (KA ) = 40. Altogether we have m2 (P/IC ) = 48 > 42 = 40 + 2 = m1 (KA ) + 2 and 1 · 32 · 48 − 171 2 1 · m1 (P/IC ) · m2 (P/IC ) − dimK (KA ). = 2 Hence this example illustrates that the assumption deg C = 512 < 597 =

m2 (P/IC ) ≤ m1 (KA ) + 2 in Theorem 3.3 is necessary and can not be dropped. Example 3.6. We take a (non-reduced) curve C ⊂ P3 defined by the ideal IC := (x0 , x1 )12 + (f ), where f ∈ (x0 , x1 ) is a generic form of degree 15 (cf. also [5, Example 2.8]). This can be realized in CoCoA via the following instructions:

THE DEFICIENCY MODULE OF A CURVE AND ITS SUBMODULES

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Use P::=Q[x[0..3]]; J:=Ideal(x[0],x[1])^12; F:=x[0]*GenForm(14,50)+x[1]*GenForm(14,50); IC:=J+Ideal(F); Here the procedure GenForm(D,N) returns a general form of degree D with randomized coefficients in the interval [−N, N ]: Define GenForm(D,N) V:=Gens(Ideal(Indets())^D); Return Sum([Rand(-N,N)*V[I] | I In 1..Len(V)]); EndDefine; If we compute the Hilbert polynomial of C with Hilbert(P/IC); we get HPC (t) = HPP/IC (t) = 12t + 870 for t ≥ 24, i.e. deg(C) = 12. Furthermore, the computation of the minimal graded free resolution of P/IC via Res(P/IC); yields 0 --> P^11(-27) --> P^12(-13)(+)P^12(-26) --> P^13(-12)(+)P(-15) --> P ------------------------------Hence m1 (P/IC ) = 12 and m2 (P/IC ) = 13. We obtain the deficiency module M (C) and its submodule KA via MC:=KDual(DeficiencyDual(IC)); KA:=K_A(MC); The dimension of the deficiency module can be calculated, for example, as the length of a normal basis defined in Section 1. The command Len(NormalBasisM(MC)); then gives dimK M (C) = 56056. Likewise, Len(NormalBasisM(KA)); gives the codimension of KA in M (C), which yields dimK KA = 66. The problem of determining m1 (KA ), i.e. the degree of a minimal generator of KA , is more delicate, since degree shifts are not supported in the current version of CoCoA. We work around this as follows: Looking at the resolution of IC , we find that Ext3 (P/IC , P ) ∼ = P 11 (27)/N . Calculating the Hilbert function via

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GUNTRAM HAINKE AND ALMAR KAID

Hilbert(DeficiencyDual(IC)); we get nontrivial terms in degrees 0 to 166, i.e. Ext3 (P/IC , P ) is concentrated in degrees −27 to 139. The K-dual then is concentrated in degrees −139 to 27, and taking the degree shift into account as given in Theorem 1.5, we see that the last non-trivial component of the deficiency module is in degree 23. Calculating the Hilbert function of KA with the help of HilbertSeriesShifts we find that the Hilbert function of KA is of the shape HFKA : 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0. Taking the various shifts and twists into account, this has to be interpreted as HFKA (t) = t − 12 for 13 ≤ t ≤ 23 and HFKA (t) = 0 otherwise. In particular, we get m1 (KA ) = 13. Hence the assumption m2 (P/IC ) = 13 ≤ 15 = 13 + 2 = m1 (KA ) + 2 of Theorem 3.3 holds and the degree bound given there is sharp since we have 1 1 deg(C) = 12 = · 12 · 13 − 66 = · m1 (P/IC ) · m2 (P/IC ) − dimK (KA ). 2 2 References [1] S. Beck and M. Kreuzer. How to compute the canonical module of a set of points. Algorithms in algebraic geometry and applications (Santander, 1994), volume 143 of Progr. Math., pages 51–78. Birkh¨ auser, Basel, 1996. [2] D. Eisenbud. The Geometry of Syzygies. Springer-Verlag, 2005. [3] G. Hainke and A. Kaid. Constructions of Gorenstein rings. in this volume, 2006. [4] R. Hartshorne. Algebraic Geometry. Springer, New York, 1977. [5] J. Migliore, U. Nagel, and T. R¨ omer. Extensions of the multiplicity conjecture. ArXiv, 2005. ¨ t fu ¨r Mathematik, Universita ¨ t Bielefeld, D-33501 Bielefeld, Fakulta Germany E-mail address: [email protected] Department of Pure Mathematics, University of Sheffield, Hounsfield Road, Sheffield S3 7RH, United Kingdom E-mail address: [email protected]